8.6: Islam
Central Asian had already been exposed to many foreign systems of belief prior to the coming of Islam in the 7th Century. Local merchants conducting long-distance trade along the Silk Road came into contact with many different religious doctrines. As western ideas traveled eastward and eastern concepts filtered into the west. In this manner, the Silk Road carried Buddhism, Judaism, Zoroastrianism (an Iranian religion with both monotheistic and dualist elements), Siberian Shamanism, and even Nestorian Christianity to Central Asia. By the 10th Century, however, all of the non-monotheistic faiths had disappeared from Central Asia in the aftermath of Islamic conquest.
During the 8th Century, Qutayba ibn Muslim (669 – 716) expanded the presence of Islam in Central Asia after his forces defeated the Persians. By 715 CE, the conquest of Transoxiana was complete. Later still, in 751, the Battle of Talas River secured Central Asia and repulsed the only major challenge to Islamic rule: China.
The Persian Samanids (819 – 999) made Islam the official state religion and established a school of theology in Bukhara. But Islamization did not take place overnight; instead, it took centuries. Transoxiana slowly Islamicized, though it never Arabized. The peoples in the area remained culturally Turkic and Persian. Central Asia retained its Turkic and Persian languages, albeit with a heavy Arabic influence in religious vocabulary.
Unlike other places, where the religion filtered into society from the bottom up, as was the case in Southeast Asia, Islamization in Central Asia occurred from the top down. The process incorporated native peoples, who took part in the process of conversion. Local leaders submitted to the faith in order to maintain their social status and elite position in society. The conquerors offered the Central Asian nobility important positions in the administration so long as they professed Islam.
Islam also displayed a remarkable ability to assimilate indigenous Central Asian frontier customs as it advanced through the area. This action allowed some traditional practices to remain so long as it accelerated conversion. By accepting certain harmless practices in order to Islamicize Central Asia, these Muslims mirrored Christian efforts to Christianize Eastern Europe.
Sufism
The Islam of Central Asia differed greatly from that which originated in the Arabian Peninsula. This vast region embraced Islam, remaking it culturally it's own. Essentially, Sufism emerged from the fusion of Islamic sedentary civilization and Buddhist nomadic culture.
Central Asians were generally receptive and tolerant of foreign beliefs. For example, missionaries of other religions had to make concessions to the native culture. The urban-dwelling and agricultural populations of the region generally accepted Sunni Islam and the law of the Sharia. However, the culture and lifestyle of the itinerant peoples of the steppe did not readily conform to the rigors of Islamic law. Sufism helped convert these tribes to Islam, in part due to its doctrinal flexibility. What arose from the mix of orthodox Islam and Turkic pastoral nomadism was a uniquely Central Asian brand of Sufism.
Sufi merchants were largely responsible for bringing Islam to the region. Central Asian Sufi orders such as the Yasaviyah established themselves along trade routes in order to reach out to travelers. Meanwhile, the Naqshbandi Order , operating in travel lodges, spread Sufism by ministering to Iranian and Tajik peoples.


14.3: Adding Information to Graphs
a graph in which each edge is assigned a weight or cost, usually a numerical value
an ordered triple \((V,E,w)\text{,}\) where \((V,E)\) is an ordinary graph and \(w: E \rightarrow W\) is a function with some set \(W\) as codomain
elements of the image \(w(E) \subseteq W\)
We usually label each edge with its weight on diagrams for the graph.
A road map with distances as weights is a weighted graph. Below is a simplified road map of the area around Camrose, Alberta. The vertex set is the set of cities, and the edge set is the set of highways. For example, the two-city set {Camrose, Edmonton} represents the edge on the graph between Camrose and Edmonton, and corresponds to Highway 21.
| \(e\) | \(w(e)\) |
| {Camrose, Edmonton} | \(94\) |
| {Edmonton, Leduc} | \(35\) |
| {Leduc, Wetaskiwin} | \(35\) |
| {Wetaskiwin, Camrose} | \(40\) |
The edges in the graph are weighted by the (rounded) highway distances between cities. Formally, this is a function \(w\) from the edge set to the natural numbers. The input-output relationship defining this function is tabulated above right.
Variations on Example \(\PageIndex{1}\) include any kind of transportation or communication network with transportation/communication lines as edges. Possible weights assigned to an edge include: length of the line; amount of time it takes a vehicle/message to travel along the line from one node to the next; capacity of the line in vehicles/passengers/messages/data per unit time; etc..
a graph in which each edge can be given a direction, “pointing” to one of its incident vertices
an ordered pair \((V,E)\text{,}\) where \(V\) is a finite set and \(E\) is an unordered, possibly empty list of elements of \(V \times V\)
Again, elements of \(V\) are the vertices and elements of \(E\) are the edges of the graph. For an ordinary graph, edges were represented by subsets of \(V\) because when specifying an edge, the order of the vertices which are to be incident to the edge is irrelevant. For a directed graph, the order of the vertices incident to an edge now matters, so we use ordered pairs of vertices to specify an edge. If \(e = (v,v') \in E\) for some \(v,v' \in V\text{,}\) consider the direction of \(e\) to be \(v \to v'\text{.}\)
Consider
\begin{gather*} V = \{ v_1, v_2, v_3, v_4 \}, \\ E = \{ (v_1,v_2), (v_1,v_2), (v_2,v_3), (v_3,v_2), (v_4,v_3), (v_4,v_4) \}. \end{gather*}
We draw the graph \(G = (V,E)\) with arrows to indicate the direction of edges.
Invent a formal definition for directed, weighted graph .


8.1: The Sampling Process
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The sampling process comprises of several stage. The first stage is defining the target population. A population can be defined as all people or items ( unit of analysis ) with the characteristics that one wishes to study. The unit of analysis may be a person, group, organization, country, object, or any other entity that you wish to draw scientific inferences about. Sometimes the population is obvious. For example, if a manufacturer wants to determine whether finished goods manufactured at a production line meets certain quality requirements or must be scrapped and reworked, then the population consists of the entire set of finished goods manufactured at that production facility. At other times, the target population may be a little harder to understand. If you wish to identify the primary drivers of academic learning among high school students, then what is your target population: high school students, their teachers, school principals, or parents? The right answer in this case is high school students, because you are interested in their performance, not the performance of their teachers, parents, or schools. Likewise, if you wish to analyze the behavior of roulette wheels to identify biased wheels, your population of interest is not different observations from a single roulette wheel, but different roulette wheels (i.e., their behavior over an infinite set of wheels).
The second step in the sampling process is to choose a sampling frame . This is an accessible section of the target population (usually a list with contact information) from where a sample can be drawn. If your target population is professional employees at work, because you cannot access all professional employees around the world, a more realistic sampling frame will be employee lists of one or two local companies that are willing to participate in your study. If your target population is organizations, then the Fortune 500 list of firms or the Standard & Poor’s (S&P) list of firms registered with the New York Stock exchange may be acceptable sampling frames.
Note that sampling frames may not entirely be representative of the population at large, and if so, inferences derived by such a sample may not be generalizable to the population. For instance, if your target population is organizational employees at large (e.g., you wish to study employee self-esteem in this population) and your sampling frame is employees at automotive companies in the American Midwest, findings from such groups may not even be generalizable to the American workforce at large, let alone the global workplace. This is because the American auto industry has been under severe competitive pressures for the last 50 years and has seen numerous episodes of reorganization and downsizing, possibly resulting in low employee morale and self-esteem. Furthermore, the majority of the American workforce is employed in service industries or in small businesses, and not in automotive industry. Hence, a sample of American auto industry employees is not particularly representative of the American workforce. Likewise, the Fortune 500 list includes the 500 largest American enterprises, which is not representative of all American firms in general, most of which are medium and smallsized firms rather than large firms, and is therefore, a biased sampling frame. In contrast, the S&P list will allow you to select large, medium, and/or small companies, depending on whether you use the S&P large-cap, mid-cap, or small-cap lists, but includes publicly traded firms (and not private firms) and hence still biased. Also note that the population from which a sample is drawn may not necessarily be the same as the population about which we actually want information. For example, if a researcher wants to the success rate of a new “quit smoking” program, then the target population is the universe of smokers who had access to this program, which may be an unknown population. Hence, the researcher may sample patients arriving at a local medical facility for smoking cessation treatment, some of whom may not have had exposure to this particular “quit smoking” program, in which case, the sampling frame does not correspond to the population of interest.
The last step in sampling is choosing a sample from the sampling frame using a well-defined sampling technique. Sampling techniques can be grouped into two broad categories: probability (random) sampling and non-probability sampling. Probability sampling is ideal if generalizability of results is important for your study, but there may be unique circumstances where non-probability sampling can also be justified. These techniques are discussed in the next two sections.


41.11: Human Osmoregulatory and Excretory Systems - Nephron- The Functional Unit of the Kidney
- Explain the role of the nephron as the functional unit of the kidney
The nephron, the functional unit of the kidney, is responsible for removing waste from the body. Each kidney is composed of over one million nephrons that dot the renal cortex, giving it a granular appearance when sectioned sagittally (from front to rear). Eighty-five percent of nephrons are cortical nephrons, deep in the renal cortex; the remaining 15 percent are juxtamedullary nephrons, which lie in the renal cortex close to the renal medulla.
Diagram of a nephron
The nephron is the functional unit of the kidney. The glomerulus and convoluted tubules of the nephron are located in the cortex of the kidney, while the collecting ducts are located in the pyramids of the kidney’s medulla.
A nephron consists of three parts: a renal corpuscle, a renal tubule, and the associated capillary network, which originates from the cortical radiate arteries. The renal corpuscle, located in the renal cortex, is composed of a network of capillaries known as the glomerulus, as well as a cup-shaped chamber that surrounds it: the glomerular or Bowman’s capsule.
The renal tubule is a long, convoluted structure that emerges from the glomerulus. It can be divided into three parts based on function. The first part is called the proximal convoluted tubule (PCT), due to its proximity to the glomerulus. The second part is called the loop of Henle, or nephritic loop, because it forms a loop (with descending and ascending limbs) that goes through the renal medulla. The third part of the renal tubule is called the distal convoluted tubule (DCT); this part is also restricted to the renal cortex. This last part of the nephron connects with and empties its filtrate into collecting ducts that line the medullary pyramids. The collecting ducts amass contents from multiple nephrons, fusing together as they enter the papillae of the renal medulla.
As urine travels down the collecting duct system, it passes by the medullary interstitium, which has a high sodium concentration as a result of the loop of Henle’s countercurrent multiplier system. Urine leaves the medullary collecting ducts through the renal papillae, emptying into the renal calyces, the renal pelvis, and finally into the bladder via the ureter.
Key Points
- Kidneys contain two types of nephrons, each located in different parts of the renal cortex: cortical nephrons and juxtamedullary nephrons.
- A nephron comprises a renal corpuscle, a renal tubule, and the associated capillary network.
- Internally, kidneys are mainly composed of over one million nephrons and an extensive network of blood vessels and capillaries.
Key Terms
- glomerulus : A small intertwined group of capillaries within a kidney’s nephron that filters the blood to make urine.
- loop of Henle : A structure in a kidney’s nephron that connects the proximal convoluted tubule to the distal convoluted tubule.


16.2: Pythagorean Theorem
Here is an analog of the Pythagorean theorems ( Theorem 6.2.1 and Theorem 13..1 ) in spherical geometry.
Let \(\triangle_sABC\) be a spherical triangle with a right angle at \(C\) . Set \(a=BC_s\) , \(b=CA_s\) , and \(c=AB_s\) . Then
\(\cos c=\cos a \cdot \cos b.\)
- Proof
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Since the angle at \(C\) is right, we can choose the coordinates in \(\mathbb{R}^3\) so that \(v_C\z=(0,0,1)\) , \(v_A\) lies in the \(xz\) -plane, so \(v_A=(x_A,0,z_A)\) , and \(v_B\) lies in \(yz\) -plane, so \(v_B=(0,y_B,z_B)\) .
Applying, 16.2.3, we get that
\(\begin{aligned} z_A&=\langle v_C,v_A\rangle =\cos b, \\ z_B&=\langle v_C,v_B\rangle =\cos a.\end{aligned}\)
Applying, 16.2.1 and 16.2.3, we get that
\(\begin{aligned} \cos c &=\langle v_A,v_B\rangle= \\ &=x_A\cdot 0+0\cdot y_B+z_A\cdot z_B= \\ &=\cos b\cdot\cos a.\end{aligned}\)
In the proof, we will use the notion of the scalar product which we are about to discuss.
Let \(v_A=(x_A,y_A,z_A)\) and \(v_B=(x_B,y_B,z_B)\) denote the position vectors of points \(A\) and \(B\) . The scalar product of the two vectors \(v_A\) and \(v_B\) in \(\mathbb{R}^3\) is defined as
\[\langle v_A,v_B\rangle := x_A\cdot x_B+y_A\cdot y_B+z_A\cdot z_B.\]
Assume both vectors \(v_A\) and \(v_B\) are nonzero; suppose that \(\phi\) denotes the angle measure between them. Then the scalar product can be expressed the following way:
\[\langle v_A,v_B\rangle=|v_A|\cdot|v_B|\cdot\cos\phi, \]
where
\(\begin{aligned} |v_A|&=\sqrt{x_A^2+y_A^2+z_A^2}, & |v_B|&=\sqrt{x_B^2+y_B^2+z_B^2}.\end{aligned}\)
Now, assume that the points \(A\) and \(B\) lie on the unit sphere \(\Sigma\) in \(\mathbb{R}^3\) centered at the origin. In this case \(|v_A|=|v_B|=1\) . By 16.2.2 we get that
\[\cos AB_s=\langle v_A,v_B\rangle.\]
AShow that if \(\triangle_s ABC\) is a spherical triangle with a right angle at \(C\) , and \(AC_s=BC_s=\dfrac{\pi}{4}\) , then \(AB_s=\dfrac{\pi}{3}\) .
- Hint
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Applying the Pythagorean theorem, we get that
\(\cos AB_s = \cos AC_s \cdot \cos BC_s = \dfrac{1}{2}.\)
Therefore, \(AB_s = \dfrac{\pi}{3}.\)
Alternatively, look at the tessellation of a hemisphere on the picture; it is made from 12 copies of \(\triangle_s ABC\) and yet 4 equilateral spherical triangles. From the symmetry of this tessellation, it follows that \([AB]_s\) occupies \(\dfrac{1}{6}\) of the equator; that is, \(AB_s = \dfrac{\pi}{3}\).


13.3: Internet of Things (IoT)
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- Ly-Huong T. Pham, Tejal Desai-Naik, Laurie Hammond, & Wael Abdeljabbar
- ASCCC Open Educational Resources Initiative (OERI)
Internet of Things (IoT)
Rouse (2019) explains that IoT is implemented as a set of web-enabled physical objects or things embedded with software, hardware, sensors, processors to collect and send data as they acquire from their environments. A ‘thing’ could be just about anything, a machine, an object, an animal, or even people as long as each thing has an embedded unique ID and is web-enabled.
In a report by McKinsey & Company on the Internet of Things (Chui et al., 2010), six broad applications are identified:
- Tracking behavior . When products are embedded with sensors, companies can track these products' movements and even monitor interactions with them. Business models can be fine-tuned to take advantage of this behavioral data. Some insurance companies, for example, are offering to install location sensors in customers’ cars. That allows these companies to base the price of policies on how a car is driven and where it travels.
- Enhanced situational awareness . Data from large numbers of sensors, for example, in infrastructure (such as roads and buildings), or to report on environmental conditions (including soil moisture, ocean currents, or weather), can give decision-makers a heightened awareness of real-time events, particularly when the sensors are used with advanced display or visualization technologies. Security personnel, for instance, can use sensor networks that combine video, audio, and vibration detectors to spot unauthorized individuals who enter restricted areas.
- Sensor-driven decision analysis . The Internet of Things also can support longer-range, more complex human planning and decision making. The technology requirements – tremendous storage and computing resources linked with advanced software systems that generate various graphical displays for analyzing data – rise accordingly.
- Process optimization . Some industries, such as chemical production, are installing legions of sensors to bring much greater granularity to monitoring. These sensors feed data to computers, which in turn analyze the data and then send signals to actuators that adjust processes – for example, by modifying ingredient mixtures, temperatures, or pressures.
- Optimized resource consumption . Networked sensors and automated feedback mechanisms can change usage patterns for scarce resources, such as energy and water. This can be accomplished by dynamically changing the price of these goods to increase or reduce demand.
- Complex autonomous systems. The most demanding use of the Internet of Things involves the rapid, real-time sensing of unpredictable conditions and instantaneous responses guided by automated systems. This kind of machine decision-making mimics human reactions, though at vastly enhanced performance levels. The automobile industry, for instance, is stepping up the development of systems that can detect imminent collisions and take evasive action.
IoT has evolved since the 1970s, and by 2020, it is now most associated with smart homes. Products such as smart thermostats, smart doors, lights, home security systems, home appliances, etc. For example, Amazon Echo, Google Home, Apple’s HomePod are smart home hubs to manage all the smart IoT in the home. More and more IoT devices will continue to be offered as vendors seek to make everything ‘smart.’
Autonomous
A trend that is emerging is autonomous robots and vehicles. By combining software, sensors, and location technologies, devices that can operate themselves to perform specific functions are being developed. These take the form of creations such as medical nanotechnology robots (nanobots), self-driving cars, self-driving trucks, drones, or crewless aerial vehicles (UAVs).
 While still an emerging field, it is showing promise for applications in the medical field. For example, a set of nanobots could be introduced into the human body to combat cancer or a specific disease. In March of 2012, Google introduced the world to their driverless car by releasing a video on YouTube showing a blind man driving the car around the San Francisco area (or search for "Self-Driving Car Test: Steve Mahan). The car combines several technologies, including a laser radar system, worth about $150,000.
By 2020, 38 states have enacted some legislation allowing various activities from conducting studies, limited pilot testing, full deployment of commercial motor vehicles without a human operator; The details can be found at ghsa.org .
The Society of Automotive Engineers ( SAE, 2018 ) has designed a zero to five rating system detailing the varying levels of automation — the higher the level, the more automated the vehicle is.
- Level Zero: No Automation – The driver does all the driving without any help from the vehicle
- Level One: Driver Assistance – The vehicle helps steer or speed up/slow down, but the driver still does the driving.
- Level Two: Partial Automation – The vehicle helps with one or more systems, but the driver still does the driving.
- Level Three: Conditional Automation – The vehicle helps with steering and brake/acceleration, but the driver still needs to monitor, can intervene as necessary still sitting in the driver seat.
- Level Four: High Automation – The vehicle completes all driving duties even if the driver does not intervene in limited conditions (i.e., local taxis)
- Level Five: Full Automation – The vehicle completes all duties without a driver on all roads in all conditions.
Consumers have begun seeing the features in levels 1 and 3 being integrated with today’s non-autonomous cars, and this trend is expected to continue.
A UAV often referred to as a “drone,” is a small airplane or helicopter that can fly without a pilot. Instead of a pilot, they are either run autonomously by computers in the vehicle or operated by a person using a remote control. While most drones today are used for military or civil applications, there is a growing market for personal drones. For a few hundred dollars, a consumer can purchase a drone for personal use.
Commercial use of UAV is beginning to emerge. Companies such as Amazon plan to deliver their packages to customers using drones, Walmart plans to use drones to carry things in their stores. This sector is forecasted to become a $12.6B worldwide market by 2025 (Statista.com, 2019) .
References:
Autonomous Vehicles. Retrieved December 10, 2020, from https://www.ghsa.org/state-laws/issues/autonomous%20vehicles .
Chui, M. and Roberts R (2010, March 1). The Internet of Things . Retrieved December 10, 2020, from https://www.mckinsey.com/industries/technology-media-and-telecommunications/our-insights/the-internet-of-things .
Rouse, Margaret (2019). Internet of things (IoT) . IoT Agenda. Retrieved December 11, 2020, from https://internetofthingsagenda.techtarget.com/definit ion/Internet-of-Things-IoT .
SAE International Releases Updated Visual Chart for Its “Levels of Driving Automation” Standard for Self-Driving Vehicles (2018). Retrieved December 10, 2020, from https://www.sae.org/news/press-room/2018/12/sae-international-releases-updated-visual-chart-for-its-%E2%80%9Clevels-of-driving-automation%E2%80%9D-standard-for-self-driving-vehicles .
Statista. Commercial Drones are Taking Off (2019). Retrieved December 11, 2020, from https://www.statista.com/chart/17201/commecial-drones-projected-growth/ .


 In tonal music, cadences are classified by their chord progressions.
A cadence is any place in a piece of music that has the feel of an ending point. This can be either a strong, definite stopping point - the end of the piece, for example, or the end of a movement or a verse - but it also refers to the "temporary-resting-place" pauses that round off the ends of musical ideas within each larger section.
A musical phrase , like a sentence, usually contains an understandable idea, and then pauses before the next idea starts. Some of these musical pauses are simply take-a-breath-type pauses, and don't really give an "ending" feeling. In fact, like questions that need answers, many phrases leave the listener with a strong expectation of hearing the next, "answering", phrase. Other phrases, though, end with a more definite "we've arrived where we were going" feeling. The composer's expert control over such feelings of expectation and arrival are one of the main sources of the listener's enjoyment of the music.
Like a story, a piece of music can come to an end by simply stopping, but most listeners will react to such abruptness with dissatisfaction: the story or music simply "stopped" instead of "ending" properly. A more satisfying ending, in both stories and music, is usually provided by giving clues that an end is coming, and then ending in a commonly-accepted way. Stories are also divided into paragraphs, chapters, stanzas, scenes, or episodes, each with their own endings, to help us keep track of things and understand what is going on. Music also groups phrases and motifs into verses, choruses, sections, and movements, marked off by strong cadences to help us keep track of them. In good stories, there are clues in the plot and the pacing - in the Western tradition, the chase gets more exciting, characters good and bad get what they deserve, the inevitable tragedy occurs, or misunderstandings get resolved - that signal that the end of the story is nearing. Similarly, in music there are clues that signal to the listener that the end is coming up. These clues may be in the form ; in the development of the musical ideas; in the music's tempo , texture , or rhythmic complexity; in the chord progression ; even in the number and length of the phrases (Western listeners are fond of powers of two ). Like the ending of a story, an ending in music is more satisfying if it follows certain customs that the listener expects to hear. If you have grown up listening to a particular musical tradition, you will automatically have these expectations for a piece of music, even if you are not aware of having them. And like the customs for storytelling, these expectations can be different in different musical traditions.
Some things that produce a feeling of cadence
- Harmony - In most Western and Western-influenced music (including jazz and "world" musics), harmony is by far the most important signal of cadence. One of the most fundamental "rules" of the major-minor harmony system is that music ends on the tonic . A tonal piece of music will almost certainly end on the tonic chord, although individual phrases or sections may end on a different chord (the dominant is a popular choice). But a composer cannot just throw in a tonic chord and expect it to sound like an ending; the harmony must "lead up to" the ending and make it feel inevitable (just as a good story makes the ending feel inevitable, even if it's a surprise). So the term cadence, in tonal music, usually refers to the "ending" chord plus the short chord progression that led up to it. There are many different terms in use for the most common tonal cadences; you will find the most common terms below . Some (but not all) modal musics also use harmony to indicate cadence, but the cadences used can be quite different from those in tonal harmony.
- Melody - In the major/minor tradition, the melody will normally end on some note of the tonic chord triad , and a melody ending on the tonic will give a stronger (more final-sounding) cadence than one ending on the third or fifth of the chord. In some modal musics, the melody plays the most important role in the cadence. Like a scale, each mode also has a home note, where the melody is expected to end. A mode often also has a formula that the melody usually uses to arrive at the ending note. For example, it may be typical of one mode to go to the final note from the note one whole tone below it; whereas in another mode the penultimate note may be a minor third above the final note. (Or a mode may have more than one possible melodic cadence, or its typical cadence may be more complex.)
- Rhythm - Changes in the rhythm , a break or pause in the rhythm, a change in the tempo , or a slowing of or pause in the harmonic rhythm are also commonly found at a cadence.
- Texture - Changes in the texture of the music also often accompany a cadence. For example, the music may momentarily switch from harmony to unison or from counterpoint to a simpler block-chord homophony .
- Form - Since cadences mark off phrases and sections, form and cadence are very closely connected, and the overall architecture of a piece of music will often indicate where the next cadence is going to be - every eight measures for a certain type of dance, for example.  An accomplished composer may "tease" you by seeming to lead to a cadence in the expected place, but then doing something unexpected instead.)
Harmonic analysis , form , and cadence in Western music are closely interwoven into a complex subject that can take up an entire course at the college-music-major level. Complicating matters is the fact that there are several competing systems for naming cadences. This introductory course cannot go very deeply into this subject, and so will only touch on the common terms used when referring to cadences. Unfortunately, the various naming systems may use the same terms to mean different things, so even a list of basic terms is a bit confusing.
Some Tonal Cadence Terms
- Authentic - A dominant chord followed by a tonic chord (V-I, or often V7-I).
- Complete Cadence - same as authentic cadence.
- Deceptive Cadence - This refers to times that the music seems to lead up to a cadence, but then doesn't actually land on the expected tonic, and also often does not bring the expected pause in the music. A deceptive cadence is typically in a major key, and is the dominant followed by the submediant (V-vi). This means the substituted chord is the relative minor of the tonic chord.
- False Cadence - Same as deceptive cadence.
- Full Close - Same as authentic cadence.
- Half-cadence - May refer to a cadence that ends on the dominant chord (V). This type of cadence is more common at pause-type cadences than at full-stop ones. OR may have same meaning as plagal cadence.
- Half close - Same as plagal cadence.
- Imperfect Cadence - May refer to an authentic (V-I) cadence in which the chord is not in root position, or the melody does not end on the tonic. OR may mean a cadence that ends on the dominant chord (same as one meaning of half-cadence).
- Interrupted Cadence - Same as deceptive cadence.
- Perfect Cadence - Same as authentic cadence. As its name suggests, this is considered the strongest, most final-sounding cadence. Some do not consider a cadence to be completely perfect unless the melody ends on the tonic and both chords (V and I) are in root position .
- Plagal Cadence - A subdominant chord followed by a tonic chord (IV-I). For many people, this cadence will be familiar as the "Amen" chords at the end of many traditional hymns.
- Semi-cadence - Same possible meanings as half cadence.
You can listen to a few simple cadences here: Perfect Cadence , Plagal Cadence , Half-cadence , Deceptive Cadence . The figure below also shows some very simple forms of some common cadences. The first step in becoming comfortable with cadences is to start identifying them in music that is very familiar to you. Find the pauses and stops in the music. Do a harmonic analysis of the last few chords before each stop, and identify what type of cadence it is. Then see if you can begin to recognize the type of cadence just by listening to the music.
Examples of Common Cadences
1 (a): Perfect Cadence in C major
1(b): Plagal Cadence in C major
1(c): Deceptive Cadence in C major
Figure 1
Identify the type of cadence in each excerpt. (Hint: First identify the key and then do a harmonic analysis of the progression.
Solution
Figure 3: Notice that the half cadence looks like (and in fact is) a modulation to the dominant. In this very common progression, the dominant seventh of the dominant (which requires an accidental) makes the dominant feel like a very strong resting point, and the piece will continue on in the dominant key for a while, before returning to the tonic key. Also notice the accidental required in the minor key to make the (major) dominant chord.


Page 1.2: Real Numbers
Although some of the terminology might be new, the ideas presented in this section should be somewhat familiar to us.
For us, our entire study of calculus will deal with real numbers . In order to rigorously define what we mean by a real number, a lot of machinery that is beyond the scope of our calculus courses are required and so we will rely on our intuition to understand real numbers. Allow us to review some types of numbers.
Definition: The natural numbers are defined to be the set containing the numbers \(1, 2, 3, 4, \ldots\). We denote the set of natural numbers by \( \mathbb{N} \). So \( \mathbb{N} = \{1, 2, 3, 4, \ldots \} \).
Definition: The integers consist of the natural numbers together with the negatives and zero. We denote the set of integers by \( \mathbb{Z} \). So \( \mathbb{Z} = \{\ldots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \ldots \} \). .
Definition: The rational numbers consist of all numbers of the form \( \dfrac{m}{n} \) where \(m\) and \(n\) are integers and \(n \neq 0\). We denote the set of integers by \( \mathbb{Q} \).
The above definition is saying that a rational number is just a ratio of two integers. Here are some examples of rational numbers: \( \frac{1}{2}, \frac{9}{10}, 0.61 = \frac{61}{100}, - 0.23 = - \frac{23}{100} \).
However, not all real numbers can be expressed as a ratio of two integers. For instance, although we do not prove this here, note that \( \sqrt{2} \) cannot be expressed as a ratio of two integers. Additionally, \( \pi \) cannot be expressed as a ratio of two integers. These numbers are called irrational numbers . Putting the rational numbers and irrational numbers together yields what we call the set of real numbers which we denote by \( \mathbb{R} \).
The real numbers follow certain rules which we are familiar with. These rules are the presented below.
Suppose \(a, b\) and \(c\) are real numbers. Then the following are true:
1) \( a + b = b + a \) (this is called the commutative property of addition)
2) \( a \times b = b \times a \) (this is called the commutative property of multiplication)
3) \( (a + b) + c = a + (b + c) \) (this is called the associative property of addition)
4) \( (a \times b) \times c = a \times (b \times c) \) (this is called the associative property of multiplication)
5) \( a(b+c) = ab + ac \) (this is called the distributive property)
In words, the above list is saying when we add two numbers, the order does not matter. The same is true for multiplication. Additionally, when we add three numbers, it does not matter which two we add first. The same can be said for multiplication. And finally, property 5 is saying that when we multiply a number by a sum of two numbers, we can multiply the outside number to each inside number and add the resulting products.


7.1: Collocates
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The (orthographic) word plays a central role in corpus linguistics. As suggested in Chapter 4, this is in no small part due to the fact that all corpora, whatever additional annotations may have been added, consist of orthographically represented language. This makes it easy to retrieve word forms. Every concordancing program offers the possibility to search for a string of characters – in fact, some are limited to this kind of query.
However, the focus on words is also due to the fact that the results of corpus linguistic research quickly showed that words (individually and in groups) are more interesting and show a more complex behavior than traditional, grammar-focused theories of language assumed. An area in which this is very obvious, and which has therefore become one of the most heavily researched areas in corpus linguistics, is the way in which words combine to form so-called collocations .
This chapter is dedicated entirely to the discussion of collocation. At first, this will seem like a somewhat abrupt shift from the topics and phenomena we have discussed so far – it may not even be immediately obvious how they fit into the definition of corpus linguistics as “the investigation of linguistic research questions that have been framed in terms of the conditional distribution of linguistic phenomena in a linguistic corpus”, which was presented at the end of Chapter 2. However, a closer look will show that studying the co-occurrence of words and/ or word forms is simply a special case of precisely this kind of research program.
7.1 Collocates
Trivially, texts are not random sequences of words. There are several factors influencing the likelihood of two (or more) words occurring next to each other.
First, the co-occurrence of words in a sequence is restricted by grammatical considerations. For example, a definite article cannot be followed by another definite article or a verb, but only by a noun, by an adjective modifying a noun, by an adverb modifying such an adjective or by a post-determiner. Likewise, a transitive verb requires a direct object in the form of a noun phrase, so – barring cases where the direct object is pre- or post-posed – it will be followed by a word that can occur at the beginning of a noun phrase (such as a pronoun, a determiner, an adjective or a noun).
Second, the co-occurrence of words is restricted by semantic considerations. For example, the transitive verb drink requires a direct object referring to a liquid, so it is probable that it will be followed by words like water , beer , coffee , poison , etc., and improbable that it will be followed by words like bread , guitar , stone , democracy , etc. Such restrictions are treated as a grammatical property of words (called selection restrictions ) in some theories, but they may also be an expression of our world knowledge concerning the activity of drinking.
Finally, and related to the issue of world knowledge, the co-occurrence of words is restricted by topical considerations. Words will occur in sequences that correspond to the contents we are attempting to express, so it is probable that co-occurring content words will come from the same discourse domain.
However, it has long been noted that words are not distributed randomly even within the confines of grammar, lexical semantics, world knowledge, and communicative intent. Instead, a given word will have affinities to some words, and disaffinities to others, which we could not predict given a set of grammatical rules, a dictionary and a thought that needs to be expressed. One of the first principled discussions of this phenomenon is found in Firth (1957).   Young is much more frequently found than old . (Firth 1957: 194f)
Note that Firth, although writing well before the advent of corpus linguistics, refers explicitly to frequency as a characteristic of collocations. The possibility of using frequency as part of the definition of collocates, and thus as a way of identifying them, was quickly taken up. Halliday (1961) provides what is probably the first strictly quantitative definition (cf. also Church & Hanks (1990) for a more recent comprehensive quantitative discussion):
Collocation is the syntagmatic association of lexical items, quantifiable, textually, as the probability that there will occur, at n removes (a distance of n lexical items) from an item x, the items a, b, c... Any given item thus enters into a range of collocation, the items with which it is collocated being ranged from more to less probable... (Halliday 1961: 276)
7.1.1 Collocation as a quantitative phenomenon
Essentially, then, collocation is just a special case of the quantitative corpus linguistic research design adopted in this book: to ask whether two words form a collocation (or: are collocates of each other) is to ask whether one of these words occurs in a given position more frequently than expected by chance under the condition that the other word occurs in a structurally or sequentially related position. In other words, we can decide whether two words a and b can be regarded as collocates on the basis of a contingency table like that in Table 7.1. The FIRST POSITION in the sequence is treated as the dependent variable, with two values: the word we are interested in (here: WORD A), and all OTHER words. The SECOND POSITION is treated as the independent variable, again, with two values: the word we are interested in (here: WORD B), and all OTHER words (of course, it does not matter which word we treat as the dependent and which as the independent variable, unless our research design suggests a particular reason). 1
Table 7.1: Collocation
On the basis of such a table, we can determine the collocation status of a given word pair.  The necessary data are shown in Table 7.2: As discussed above, the dependent variable is the FIRST POSITION in the sequence, with the values SILLY and ¬SILLY (i.e., all words that are not ass ); the independent variable is the SECOND POSITION in the sequence, with the values ASS and ¬ASS.
 The difference between the observed and the expected frequencies is highly significant (χ 2 = 6033.8, df = 1, p < 0.001). Note that we are using the χ 2 test here because we are already familiar with it. However, this is not the most useful test for the purpose of identifying collocations, so we will discuss better options below.
Generally speaking, the goal of a quantitative collocation analysis is to identify, for a given word, those other words that are characteristic for its context of usage. Tables 7.1 and 7.2 present the most straightforward way of doing so: we simply compare the frequency with which two words co-occur to the frequencies with which they occur in the corpus in general. In other words, the two conditions across which we are investigating the distribution of a word are “next to a given other word” and “everywhere else”. This means that the corpus itself functions as a kind of neutral control condition, albeit a somewhat indiscriminate one: comparing the frequency of a word next to some other word to its frequency in the entire rest of the corpus is a bit like comparing an experimental group of subjects that have been given a particular treatment to a control group consisting of all other people who happen to live in the same city.
Often, we will be interested in the distribution of a word across two specific conditions – in the case of collocation, the distribution across the immediate contexts of two semantically related words.
In such cases, we can modify Table 7.1 as shown in Table 7.3 to identify the collocates that differ significantly between two words. There is no established term for such collocates, so we we will call them differential collocates here 2 (the method is based on Church et al. 1991).
 We can speak of silly game(s) or childish game(s) , but we may feel that the latter is more typical than the former. The relevant lemma frequencies to put this feeling to the test are shown in Table 7.4.
Table 7.4: Childish game vs. silly game (lemmas) in the BNC
The sequences childish game(s) and silly game(s) both occur in the BNC. Both combinations taken individually are significantly more frequent than expected (you may check this yourself using the frequencies from Table 7.4, the total lemma frequency of game in the BNC (20 627), and the total number of words in the BNC given in Table 7.2 above). The lemma sequence silly game is more frequent, which might lead us to assume that it is the stronger collocation. However, the direct comparison shows that this is due to the fact that silly is more frequent in general than childish , making the combination silly game more probable than the combination childish game even if the three words were distributed randomly. The difference between the observed and the expected frequencies suggests that childish is more strongly associated with game(s) than silly . The difference is significant (χ 2 = 6.49, df = 1, p < 0.05).
Researchers differ with respect to what types of co-occurrence they focus on when identifying collocations. Some treat co-occurrence as a purely sequential phenomenon defining collocates as words that co-occur more frequently than expected within a given span. Some researchers require a span of 1 (i.e., the words must occur directly next to each other), but many allow larger spans (five words being a relatively typical span size).
Other researchers treat co-occurrence as a structural phenomenon, i.e., they define collocates as words that co-occur more frequently than expected in two related positions in a particular grammatical structure, for example, the adjective and noun positions in noun phrases of the form [Det Adj N] or the verb and noun position in transitive verb phrases of the form [V [ NP (Det) (Adj) N]]. 3 However, instead of limiting the definition to one of these possibilities, it seems more plausible to define the term appropriately in the context of a specific research question. In the examples above, we used a purely sequential definition that simply required words to occur next to each other, paying no attention to their word-class or structural relationship; given that we were looking at adjective-noun combinations, it would certainly have been reasonable to restrict our search parameters to adjectives modifying the noun ass , regardless of whether other adjectives intervened, for example in expressions like silly old ass , which our query would have missed if they occurred in the BNC (they do not).
It should have become clear that the designs in Tables 7.1 and 7.3 are essentially variants of the general research design introduced in previous chapters and used as the foundation of defining corpus linguistics: it has two variables, POSITION 1 and POSITION 2, both of which have two values, namely WORD X VS. OTHER WORDS (or, in the case of differential collocates, WORD X VS. WORD Y). The aim is to determine whether the value WORD A is more frequent for POSITION 1 under the condition that WORD B occurs in POSITION 2 than under the condition that other words (or a particular other word) occur in POSITION 2.
7.1.2 Methodological issues in collocation research
We may occasionally be interested in an individual pair of collocates, such as silly ass , or in a small set of such pairs, such as all adjective-noun pairs with ass as the noun. However, it is much more likely that we will be interested in large sets of collocate pairs, such as all adjective-noun pairs or even all word pairs in a given corpus. This has a number of methodological consequences concerning the practicability, the statistical evaluation and the epistemological status of collocation research.
a. Practicability . In practical terms, the analysis of large numbers of potential collocations requires creating a large number of contingency tables and subjecting them to the χ 2 test or some other appropriate statistical test. This becomes implausibly time-consuming very quickly and thus needs to be automated in some way.
There are concordancing programs that offer some built-in statistical tests, but they typically restrict our options quite severely, both in terms of the tests they allow us to perform and in terms of the data on which the tests are performed. Anyone who decides to become involved in collocation research (or some of the large-scale lexical research areas described in the next chapter), should get acquainted at least with the simple options of automatizing statistical testing offered by spreadsheet applications. Better yet, they should invest a few weeks (or, in the worst case, months) to learn a scripting language like Perl, Python or R (the latter being a combination of statistical software and programming environment that is ideal for almost any task that we are likely to come across as corpus linguists).
b. Statistical evaluation . In statistical terms, the analysis of large numbers of potential collocations requires us to keep in mind that we are now performing multiple significance tests on the same set of data. This means that we must adjust our significance levels. Think back to the example of coin-flipping: the probability of getting a series of one head and nine tails is 0.009765. If we flip a coin ten times and get this result, we could thus reject the null hypothesis with a probability of error of 0.010744, i.e., around 1 percent (because we would have to add the probability of getting ten tails, 0.000976). This is well below the level required to claim statistical significance. However, if we perform one hundred series of ten coin-flips and one of these series consists of one head and nine tails (or ten tails), we could not reject the null hypothesis with the same confidence, as a probability of 0.010744 means that we would expect one such series to occur by chance. This is not a problem as long as we do not accord this one result out of a hundred any special importance. However, if we were to identify a set of 100 collocations with p -values of 0.001 in a corpus, we are potentially treating all of them as important, even though it is very probable that at least one of them reached this level of significance by chance.
To avoid this, we have to correct our levels of significance when performing multiple tests on the same set of data. As discussed in Section 6.6.1 above, the simplest way to do this is the Bonferroni correction, which consists in dividing the conventionally agreed-upon significance levels by the number of tests we are performing. As noted in Section 6.6.1, this is an extremely conservative correction that might make it quite difficult for any given collocation to reach significance.
Of course, the question is how important the role of p -values is in a design where our main aim is to identify collocates and order them in terms of their collocation strength. I will turn to this point presently, but before I do so, let us discuss the third of the three consequences of large-scale testing for collocation, the methodological one.
c. Epistemological considerations . We have, up to this point, presented a very narrow view of the scientific process based (in a general way) on the Popperian research cycle where we formulate a research hypothesis and then test it (either directly, by looking for counterexamples, or, more commonly, by attempting to reject the corresponding null hypothesis). This is called the deductive method. However, as briefly discussed in Chapter 3, there is an alternative approach to scientific research that does not start with a hypothesis, but rather with general questions like “Do relationships exist between the constructs in my data?” and “If so, what are those relationships?”. The research then consists in applying statistical procedures to large amounts of data and examining the results for interesting patterns. As electronic storage and computing power have become cheaper and more widely accessible, this approach – the exploratory or inductive approach – has become increasingly popular in all branches of science, particularly the social sciences. It would be surprising if corpus linguistics was an exception, and indeed, it is not. Especially the area of collocational research is typically exploratory.
In principle, there is nothing wrong with exploratory research – on the contrary, it would be unreasonable not to make use of the large amounts of language data and the vast computing power that has become available and accessible over the last thirty years. In fact, it is sometimes difficult to imagine a plausible hypothesis for collocational research projects. What hypothesis would we formulate before identifying all collocations in the LOB or some specialized corpus (e.g., a corpus of business correspondence, a corpus of flight-control communication or a corpus of learner language)? 4 Despite this, it is clear that the results of such a collocation analysis yield interesting data, both for practical purposes (building dictionaries or teaching materials for business English or aviation English, extracting terminology for the purpose of standardization, training natural-language processing systems) and for theoretical purposes (insights into the nature of situational language variation or even the nature of language in general).
But there is a danger, too: Most statistical procedures will produce some statistically significant result if we apply them to a large enough data set, and collocational methods certainly will. Unless we are interested exclusively in description, the crucial question is whether these results are meaningful. If we start with a hypothesis, we are restricted in our interpretation of the data by the need to relate our data to this hypothesis. If we do not start with a hypothesis, we can interpret our results without any restrictions, which, given the human propensity to see patterns everywhere, may lead to somewhat arbitrary post-hoc interpretations that could easily be changed, even reversed, if the results had been different and that therefore tell us very little about the phenomenon under investigation or language in general. Thus, it is probably a good idea to formulate at least some general expectations before doing a large-scale collocation analysis.
Even if we do start out with general expectations or even with a specific hypothesis, we will often discover additional facts about our phenomenon that go beyond what is relevant in the context of our original research question.  Instead, frequent adjectival collocates (ignoring second-placed wild , which exclusively refers to actual donkeys), are pompous and bad . Pompous does not really fit with the semantics that Firth’s adjectives suggest and could indicate that a semantic shift from ‘stupidity’ to ‘self-importance’ may have taken place between 1957 and 1991 (when the BNC was assembled).
This is, of course, a new hypothesis that can (and must) be investigated by comparing data from the 1950s and the 1990s.  However, it is crucial to treat this as a hypothesis rather than a result.  In order to be tested, these ideas – and any ideas derived from an exploratory data analysis – have to be turned into testable hypotheses and the constructs involved have to be operationalized. Crucially, they must be tested on a new data set – if we were to circularly test them on the same data that they were derived from, we would obviously find them confirmed.
7.1.3 Effect sizes for collocations
As mentioned above, significance testing (while not without its uses) is not necessarily our primary concern when investigating collocations. Instead, researchers frequently need a way of assessing the strength of the association between two (or more) words, or, put differently, the effect size of their co-occurrence (recall from Chapter 6 that significance and effect size are not the same). A wide range of such association measures has been proposed and investigated. They are typically calculated on the basis of (some or all) the information contained in contingency tables like those in Tables 7.1 and 7.3 above.
Let us look at some of the most popular and/or most useful of these measures. I will represent the formulas with reference to the table in Table 7.5, i.e, O 11 means the observed frequency of the top left cell, E 11 its expected frequency, R 1 the first row total, C 2 the second column total, and so on. Note that second column would be labeled OTHER WORDS in the case of normal collocations, and WORD C in the case of differential collocations. The association measures can be applied to both kinds of design.
Table 7.5: A generic 2-by-2 table for collocation research
Now all we need is a good example to demonstrate the calculations. Let us use the adjective-noun sequence good example from the LOB corpus (but horse lovers need not fear, we will return to equine animals and their properties below).
Table 7.6: Co-occurrence of good and example in the LOB
Measures of collocation strength differ with respect to the data needed to calculate them, their computational intensiveness and, crucially, the quality of their results. In particular, many measures, notably the ones easy to calculate, have a problem with rare collocations, especially if the individual words of which they consist are also rare. After we have introduced the measures, we will therefore compare their performance with a particular focus on the way in which they deal (or fail to deal) with such rare events.
<IP_ADDRESS> Chi-square
The first association measure is an old acquaintance: the chi-square statistic, which we used extensively in Chapter 6 and in Section 7.1.1 above. I will not demonstrate it again, but the chi-square value for Table 7.6 would be 378.95 (at 1 degree of freedom this means that p < 0.001, but we are not concerned with p -values here).
Recall that the chi-square test statistic is not an effect size, but that it needs to be divided by the table total to turn it into one. As long as we are deriving all our collocation data from the same corpus, this will not make a difference, since the table total will always be the same. However, this is not always the case. Where table sizes differ, we might consider using the phi value instead. I am not aware of any research using phi as an association measure, and in fact the chi-square statistic itself is not used widely either. This is because it has a serious problem: recall that it cannot be applied if more than 20 percent of the cells of the contingency table contain expected frequencies smaller than 5 (in the case of collocates, this means not even one out of the four cells of the 2-by-2 table). One reason for this is that it dramatically overestimates the effect size and significance of such events, and of rare events in general. Since collocations are often relatively rare events, this makes the chi-square statistic a bad choice as an association measure.
<IP_ADDRESS> Mutual Information
Mutual information is one of the oldest collocation measures, frequently used in computational linguistics and often implemented in collocation software. It is given in (1) in a version based on Church & Hanks (1990): 5
(1)
Applying the formula to our table, we get the following:
In our case, we are looking at cases where WORD A and WORD B occur directly next to each other, i.e., the span size is 1. When looking at a larger span (which is often done in collocation research), the probability of encountering a particular collocate increases, because there are more slots that it could potentially occur in. The MI statistic can be adjusted for larger span sizes as follows (where S is the span size):
(2)
The mutual information measure suffers from the same problem as the χ 2 statistic: it overestimates the importance of rare events. Since it is still fairly widespread in collocational research, we may nevertheless need it in situations where we want to compare our own data to the results of published studies. However, note that there are versions of the MI measure that will give different results, so we need to make sure we are using the same version as the study we are comparing our results to. But unless there is a pressing reason, we should not use mutual information at all.
<IP_ADDRESS> The log-likelihood ratio test
The G value of the log-likelihood ratio test is one of the most popular – perhaps the most popular – association measure in collocational research, found in many of the central studies in the field and often implemented in collocation software. The following is a frequently found form (Read & Cressie 1988: 134):
(3)
In order to calculate the G measure, we calculate for each cell the natural logarithm of the observed frequency divided by the expected frequency and multiply it by the observed frequency. We then add up the results for all four cells and multiply the result by two. Note that if the observed frequency of a given cell is zero, the expression O i / Ei will, of course, also be zero. Since the logarithm of zero is undefined, this would result in an error in the calculation. Thus, log(0) is simply defined as zero when applying the formula in (3).
Applying the formula in (3) to the data in Table 7.6, we get the following:
The G value has long been known to be more reliable than the χ 2 test when dealing with small samples and small expected frequencies (Read & Cressie 1988: 134ff). This led Dunning (1993) to propose it as an association measure specifically to avoid the overestimation of rare events that plagues the χ 2 test, mutual information and other measures.
<IP_ADDRESS> Minimum Sensitivity
Minimum sensitivity was proposed by Pedersen (1998) as potentially useful measure especially for the identification of associations between content words:
(4)
We simply divide the observed frequency of a collocation by the frequency of the first word (R 1 ) and of the second word (C 1 ) and use the smaller of the two as the association measure. For the data in Table 7.6, this gives us the following:
In addition to being extremely simple to calculate, it has the advantage of ranging from zero (words never occur together) to 1 (words always occur together); it was also argued by Wiechmann (2008) to correlate best with reading time data when applied to combinations of words and grammatical constructions (see Chapter 8). However, it also tends to overestimate the importance of rare collocations.
<IP_ADDRESS> Fisher’s exact test
isher’s exact test was already mentioned in passing in Chapter 6 as an alternative to the χ 2 test that calculates the probability of error directly by adding up the probability of the observed distribution and all distributions that deviate from the null hypothesis further in the same direction. Pedersen (1996) suggests using this p -value as a measure of association because it does not make any assumptions about normality and is even better at dealing with rare events than G. Stefanowitsch & Gries (2003: 238–239) add that it has the advantage of taking into account both the magnitude of the deviation from the expected frequencies and the sample size.
There are some practical disadvantages to Fisher’s exact test. First, it is computationally expensive – it cannot be calculated manually, except for very small tables, because it involves computing factorials, which become very large very quickly. For completeness’ sake, here is (one version of) the formula:
(5)
Obviously, it is not feasible to apply this formula directly to the data in Table 7.6, because we cannot realistically calculate the factorials for 236 or 836, let alone 1 011 904. But if we could, we would find that the p -value for Table 7.6 is 0.000000000001188.
Spreadsheet applications do not usually offer Fisher’s exact test, but all major statistics applications do. However, typically, the exact p -value is not reported beyond the limit of a certain number of decimal places. This means that there is often no way of ranking the most strongly associated collocates, because their p -values are smaller than this limit. For example, there are more than 100 collocates in the LOB corpus with a Fisher’s exact p -value that is smaller than the smallest value that a standard-issue computer chip is capable of calculating, and more than 5000 collocates that have p -values that are smaller than what the standard implementation of Fisher’s exact test in the statistical software package R will deliver. Since in research on collocations we often need to rank collocations in terms of their strength, this may become a problem.
<IP_ADDRESS> A comparison of association measures
Let us see how the association measures compare using a data set of 20 potential collocations.  Table 7.7 shows the combinations and their frequencies in the BNC sorted by their raw frequency of occurrence (adjectives and nouns are shown in small caps here to stress that they are values of the variables Word A and Word B, but I will generally show them in italics in the remainder of the book in line with linguistic tradition).
All combinations are perfectly normal, grammatical adjective-noun pairs, meaningful not only in the specific context of their actual occurrence. However, I have selected them in such a way that they differ with respect to their status as potential collocations (in the sense of typical combinations of words). Some are compounds or compound like combinations ( rocking horse , Trojan horse , and, in specialist discourse, common zebra ).  Some are very conventional combinations of nouns with an adjective denoting a property specific to that noun ( prancing horse , braying donkey , galloping horse – the first of these being a conventional way of referring to the Ferrari brand mark logo).  Finally, there are a number of fully compositional combinations that make sense but do not have any special status ( caparisoned mule , new horse , old donkey , young zebra , large mule , female hinny , extinct quagga ).
Table 7.7: Some collocates of the form [ADJ N equine ] (BNC)
In addition, I have selected them to represent different types of frequency relations: some of them are (relatively) frequent, some of them very rare, for some of them the either the adjective or the noun is generally quite frequent, and for some of them neither of the two is frequent.
Table 7.8 shows the ranking of these twenty collocations by the five association measures discussed above.  Common zebra is difficult to predict – it is a conventionalized expression, but not in the general language.
Table 7.8: Comparison of selected association measures for collocates of the form [ADJ N equine ] (BNC)
All association measures fare quite well, generally speaking, with respect to the compositional expressions – these tend to occur in the lower third of all lists. Where there are exceptions, the χ 2 statistic, mutual information and minimum sensitivity rank rare cases higher than they should (e.g. caparisoned mule , extinct quagga ), while the G and the p -value of Fisher’s exact test rank frequent cases higher (e.g. galloping horse ).
 Minimum sensitivity is much better, ranking most of the conventionalized cases in the top half of the list and the non-conventionalized ones further down (with the exception of jumped-up jackass , where both the individual words and their combination are very rare). The G and the Fisher p -value fare best (with no differences in their ranking of the expressions), listing the conventionalized cases at the top and the distinctive but non-conventionalized cases in the middle.
To demonstrate the problems that very rare events can cause (especially those where both the combination and each of the two words in isolation are very rare), imagine someone had used the phrase tomfool onager once in the BNC. Since neither the adjective tomfool (a synonym of silly) nor the noun onager (the name of the donkey sub-genus Equus hemionus, also known as Asiatic or Asian wild ass ) occur in the BNC anywhere else, this would give us the distribution in Table 7.9.
Table 7.9: Fictive occurrence of tomfool onager in the BNC
Applying the formulas discussed above to this table gives us a χ 2 value of 98 364 000, an MI value of 26.55 and a minimum sensitivity value of 1, placing this (hypothetical) one-off combination at the top of the respective rankings by a wide margin. Again, the log-likelihood ratio test and Fisher’s exact test are much better, putting in eighth place on both lists ( G = 36.81, p exact = 1.02 × 10 −8 ).
Although the example is hypothetical, the problem is not. It uncovers a mathematical weakness of many commonly used association measures. From an empirical perspective, this would not necessarily be a problem, if cases like that in Table 7.9 were rare in linguistic corpora. However, they are not. The LOB corpus, for example, contains almost one thousand such cases, including some legitimate collocation candidates (like herbal brews, casus belli or sub-tropical climates ), but mostly compositional combinations ( ungraceful typography, turbaned headdress, songs-of-Britain medley ), snippets of foreign languages ( freie Blicke, l’arbre rouge, palomita blanca ) and other things that are quite clearly not what we are looking for in collocation research. All of these will occur at the top of any collocate list created using statistics like χ 2 , mutual information and minimum sensitivity. In large corpora, which are impossible to check for orthographical errors and/or errors introduced by tokenization, this list will also include hundreds of such errors (whose frequency of occurrence is low precisely because they are errors).
To sum up, when doing collocational research, we should use the best association measures available. For the time being, this is the p value of Fisher’s exact test (if we have the means to calculate it), or G (if we don’t, or if we prefer using a widely-accepted association measure). We will use G through much of the remainder of this book whenever dealing with collocations or collocation-like phenomena.
1 Note that we are using the corpus size as the table total – strictly speaking, we should be using the total number of two-word sequences (bigrams) in the corpus, which will be lower: The last word in each file of our corpus will not have a word following it, so we would have to subtract the last word of each file – i.e., the number of files in our corpus – from the total. This is unlikely to make much of a difference in most cases, but the shorter the texts in our corpus are, the larger the difference will be. For example, in a corpus of tweets, which, at the time of writing, are limited to 280 characters, it might be better to correct the total number of bigrams in the way described.
2 Gries (2003b) and Gries & Stefanowitsch (2004) use the term distinctive collocate , which has been taken up by some authors; however, many other authors use the term distinctive collocate much more broadly to refer to characteristic collocates of a word.
3 Note that such word-class specific collocations are sometimes referred to as colligations, although the term colligation usually refers to the co-occurrence of a word in the context of particular word classes, which is not the same.
4 Of course we are making the implicit assumption that there will be collocates – in a sense, this is a hypothesis, since we could conceive of models of language that would not predict their existence (we might argue, for example, that at least some versions of generative grammar constitute such models). However, even if we accept this as a hypothesis, it is typically not the one we are interested in this kind of study.
5 A logarithm with a base b of a given number x is the power to which b must be raised to produce x , so, for example, log 10 (2) = 0.30103, because 10 0.30103 = 2. Most calculators offer at the very least a choice between the natural logarithm, where the base is the number e (approx. 2.7183) and the common logarithm, where the base is the number 10; many calculators and all major spreadsheet programs offer logarithms with any base. In the formula in (1), we need the logarithm with base 2; if this is not available, we can use the natural logarithm and divide the result by the natural logarithm of 2:


1.5: Defining the Writing Process
On the surface, nothing could be simpler than writing: You sit down, you pick up a pen or open a document on your computer, and you write words. But anyone who has procrastinated or struggled with writer’s block knows that the writing process is more arduous, if not somewhat mysterious and unpredictable.
People often think of writing in terms of its end product—the email, the report, the memo, essay, or research paper, all of which result from the time and effort spent in the act of writing. In this course, however, you will be introduced to writing as the recursive process of planning, drafting, and revising.
Writing is Recursive
You will focus as much on the process of writing as you will on its end product (the writing you normally submit for feedback or a grade). Recursive means circling back; and, more often than not, the writing process will have you running in circles. You might be in the middle of your draft when you realize you need to do more brainstorming, so you return to the planning stage. Even when you have finished a draft, you may find changes you want to make to an introduction. In truth, every writer must develop his or her own process for getting the writing done, but there are some basic strategies and techniques you can adapt to make your work a little easier, more fulfilling and effective.
Developing Your Writing Process
The final product of a piece of writing is undeniably important, but the emphasis of this course is on developing a writing process that works for you. Some of you may already know what strategies and techniques assist you in your writing. You may already be familiar with prewriting techniques, such as freewriting, clustering, and listing. You may already have a regular writing practice. But the rest of you may need to discover what works through trial and error. Developing individual strategies and techniques that promote painless and compelling writing can take some time. So, be patient.
A Writer’s Process: Ali Hale
Read and examine The Writing Process by Ali Hale. Think of this document as a framework for defining the process in distinct stages: Prewriting, Writing, Revising, Editing, and Publishing. You may already be familiar with these terms. You may recall from past experiences that some resources refer to prewriting as planning and some texts refer to writing as drafting.
What is important to grasp early on is that the act of writing is more than sitting down and writing something. Please avoid the “one and done” attitude, something instructors see all too often in undergraduate writing courses. Use Hale’s essay as your starting point for defining your own process.
A Writer’s Process: Anne Lamott
In the video below, Anne Lamott, a writer of both non-fiction and fiction works, as well as the instructional novel on writing Bird by Bird: Instructions on Writing , discusses her own journey as a writer, including the obstacles she has to overcome every time she sits down to begin her creative process. She will refer to terms such as “the down draft,” “the up draft,” and “the dental draft.”
As you watch, think about how her terms, “down draft,” “up draft,” “dental draft,” work with those presented by Hale’s The Writing Process . What does Lamott mean by these terms? Can you identify with her process or with the one Hale describes? How are they related?
Also, when viewing the interview, pay careful attention to the following timeframe: 11:23 to 27:27 minutes and make a list of tips and strategies you find particularly helpful. Think about how your own writing process fits with what Hale and Lamott have to say. Is yours similar? Different? Is there any new information you have learned that you did not know before exposure to these works?


12.4: Diversity and Its Impact on Companies
- How does diversity impact companies and the workforce?
Due to trends in globalization and increasing ethnic and gender diversity, it is imperative that employers learn how to manage cultural differences and individual work attitudes. As the labor force becomes more diverse there are both opportunities and challenges to managing employees in a diverse work climate. Opportunities include gaining a competitive edge by embracing change in the marketplace and the labor force. Challenges include effectively managing employees with different attitudes, values, and beliefs, in addition to avoiding liability when leadership handles various work situations improperly.
Reaping the Advantages of Diversity
The business case for diversity introduced by Taylor Cox and Stacy Blake outlines how companies may obtain a competitive advantage by embracing workplace diversity. 98 Six opportunities that companies may receive when pursuing a strategy that values diversity include cost advantages, improved resource acquisition, greater marketing ability, system flexibility, and enhanced creativity and better problem solving (see Exhibit 12.6).
Exhibit 12.6 Managing Cultural Diversity (Attribution: Copyright Rice University, OpenStax, under CC-BY 4.0 license)
Cost Advantages
Traits such as race, gender, age, and religion are protected by federal legislation against various forms of discrimination (covered later in this chapter). Organizations that have policies and procedures in place that encourage tolerance for a work climate of diversity and protect female and minority employees and applicants from discrimination may reduce their likelihood of being sued due to workplace discrimination. Cox and Blake identify this decreased liability as an opportunity for organizations to reduce potential expenses in lawsuit damages compared to other organizations that do not have such policies in place.
Additionally, organizations with a more visible climate of diversity experience lower turnover among women and minorities compared to companies that are perceived to not value diversity. 99 Turnover costs can be substantial for companies over time, and diverse companies may ameliorate turnover by retaining their female and minority employees. Although there is also research showing that organizations that value diversity experience a higher turnover of White employees and male employees compared to companies that are less diverse, 100 some experts believe this is due to a lack of understanding of how to effectively manage diversity. Also, some research shows that Whites with a strong ethnic identity are attracted to diverse organizations similarly to non-Whites. 101
Resource Acquisition
Human capital is an important resource of organizations, and it is acquired through the knowledge, skills, and abilities of employees. Organizations perceived to value diversity attract more women and minority job applicants to hire as employees. Studies show that women and minorities have greater job-pursuit intentions and higher attraction toward organizations that promote workplace diversity in their recruitment materials compared to organizations that do not. 102 When employers attract minority applicants, their labor pool increases in size compared to organizations that are not attractive to them. As organizations attract more job candidates, the chances of hiring quality employees increases, especially for jobs that demand highly skilled labor. In summary, organizations gain a competitive advantage by enlarging their labor pool by attracting women and minorities.
Marketing
When organizations employ individuals from different backgrounds, they gain broad perspectives regarding consumer preferences of different cultures. Organizations can gain insightful knowledge and feedback from demographic markets about the products and services they provide. Additionally, organizations that value diversity enhance their reputation with the market they serve, thereby attracting new customers.
System Flexibility
When employees are placed in a culturally diverse work environment, they learn to interact effectively with individuals who possess different attitudes, values, and beliefs. Cox and Blake contend that the ability to effectively interact with individuals who differ from oneself builds cognitive flexibility , the ability to think about things differently and adapt one’s perspective. When employees possess cognitive flexibility, system flexibility develops at the organizational level. Employees learn from each other how to tolerate differences in opinions and ideas, which allows communication to flow more freely and group interaction to be more effective.
Creativity and Problem Solving
Teams from diverse backgrounds produce multiple points of view, which can lead to innovative ideas. Different perspectives lead to a greater number of choices to select from when addressing a problem or issue.
Life experience varies from person to person, sometimes based on race, age, or sex. Creativity has the opportunity to flourish when those experiences are shared. Diverse teams not only produce more alternatives, but generate a broader range of perspectives to address tasks and problems. One way in which diverse teams enhance problem-solving ability is by preventing groupthink , 103 a dysfunction in decision-making that occurs in homogeneous groups as a result of group pressures and group members’ desire for conformity and consensus. Diverse group membership prevents groupthink because individuals from varied backgrounds with different values, attitudes, and beliefs can test the assumptions and reasoning of group members’ ideas.
Aligning Diversity Programs with an Organization’s Mission and Strategic Goals
Diversity helps organizations perform best when it is aligned with a specific business strategy. For example, when companies use heterogeneous management teams that are directed by an entrepreneurial strategy focusing on innovation, the companies’ productivity increases.
When an entrepreneurial strategy is not present, however, team diversity has little effect on productivity. 104 An entrepreneurial strategy includes innovation that reflects a company’s commitment to being creative, supporting new ideas, and supporting experimentation as a way to gain a competitive advantage. In other words, managers may properly utilize the multiple perspectives that emerge from heterogeneous teams by integrating them as a resource for pursuing the overall strategy of the organization.
Using Human Resources Tools Strategically
To effectively align diversity with an organization’s strategy, the human resources function must be able to engage employees at dynamic levels. Using a strategic human resources management approach to an organization can successfully integrate diversity with the organization’s goals and objectives. 105 Strategic human resources management (SHRM) is a system of activities arranged to engage employees in a manner that assists the organization in achieving a sustainable competitive advantage. SHRM practices vertically integrate with the mission and strategy of the organization while horizontally integrating human resources activities across its functional areas. By doing so, a unique set of resources can be made available to specific to the needs of the organization. Furthermore, when human resources becomes a part of the strategic planning process instead of just providing ancillary services, improved communication, knowledge sharing, and greater synergy between decision makers can occur within the organization to improve organizational functioning.
The resource-based view of the firm has been used to support the argument for diversity because it demonstrates how a diverse workforce can create a sustainable competitive advantage for organizations. Based on the resource-based view of the firm, when companies possess resources that are rare, valuable, difficult to imitate, and non-substitutable, a sustained competitive advantage can be attained. 106 The SHRM approach assumes that human capital—the current and potential knowledge, skills, and abilities of employees—is instrumental to every organization’s success and sustainability and longevity.
If a diverse composition of employees within organizations is rare, employing minorities in positions of leadership is even rarer. One exception is Northern Trust, an investment management firm that was recently listed on Forbes magazine’s 2018 Best Employers for Diversity list. 107 Thirty-eight percent of Northern Trust’s top executives are women, which is impressive because it matches the average percentage of women in fulltime one-year MBA programs over the past five years. 108 The average for S&P 500 companies is just 27%. In addition, African Americans make up 23% of Northern Trust’s board, which also demonstrates the commitment Northern Trust has to diversity. This rare degree of diversity helps Northern Trust become an employer of choice for minorities and women. In turn, attracting minority applicants increases the labor pool available to Northern Trust and increases its ability to find good talent.
Exhibit 12.7 Bank staff watching presentation The Disability Awareness Players present to the staff at Northern Trust. (Credit: JJ’s List/ flickr/ Attribution 2.0 Generic (CC BY 2.0))
Diverse companies may capitalize on the multiple perspectives that employees from different backgrounds contribute to problem solving and idea generation. In group settings, members from collectivist cultures from Asia and South America, for example, engage with others on tasks differently than members from North America. Similarly, Asians, Blacks, and Hispanics usually act more collectively and engage more interdependently than Whites, who are generally more individualistic. More harmonious working interactions benefit group cohesion and team performance, 109 and employees can grasp better ways of doing things when there is a diverse population to learn from.
For a company to attain a sustained competitive advantage, its human resource practices must be difficult to copy or imitate. As we will see later in the chapter, companies may hold one of three perspectives on workplace diversity. The integration and learning perspective results in the best outcomes for employees and the organization. However, it is not easy to become an employer that can effectively manage diversity and avoid the challenges we learned about earlier in this chapter. Historical conditions and often-complex interplay between various organizational units over time can contribute to a company’s ability to perform effectively as a diverse organization. Best practices for targeting diverse applicants or resolving conflicts based on cultural differences between employees may occur organically and later become codified into the organizational culture. Sometimes, however, the origin of diversity practices is unknown because they arose from cooperation among different functional areas (e.g., marketing and human resources working strategically with leadership to develop recruitment ideas) that occurred so long ago that not even the company itself, let alone other companies, could replicate the process.
Diversity and Organizational Performance
Research indicates that having diversity in an organization produces mixed results for its success. Some studies show a positive relationship, some show a negative relationship, and others show no relationship between diversity and performance. Some researchers believe that although findings regarding a direct relationship between diversity and success in the marketplace may be inconsistent, the relationship may be due to other variables not taken into account.
Taking the resource-based view perspective, Richard and colleagues demonstrated that racially diverse banking institutions focused on innovation experienced greater performance than did racially diverse banks with a low focus on innovation. 110 These findings suggest that for the potential of racial diversity to be fully realized, companies should properly manage the system flexibility, creativity, and problem-solving abilities used in an innovative strategy. Other studies show that when top management includes female leadership, firm performance improves when organizations are innovation driven. 111
- What are the challenges and opportunities that diversity provides to companies?
- What are the responsibilities of human resources regarding diversity?
- Can diversity be a strategic advantage to organizations?
References
98. Cox, T.H. & Blake, S. (1991). Managing cultural diversity: Implications for organizational competitiveness. Academy of Management Executive, 5(3): 45-56.
99. Williams, K., & O'Reilly, CA. 1998. Demography and diversity: A review of 40 years of research.In B. Staw and R. Sutton (Eds.), Research in organizational behavior, 20: 77-140. Greenwich, CT: JAI Press.
100. Tsui, A.S., Egan, T. D., & O’Reilly, C.A. 1992. Being different: relational demography and organizational attachment. Administrative Science Quarterly, 37: 549-579.
101. Kim, S.S. & Gelfand, M. J. (2003).The influence of ethnic identity on perceptions of organizational recruitment. Journal of Vocational Behavior, 63: 396- 416.
102. Perkins, L. A., Thomas, K. M., & Taylor, G. A. 2000. Advertising and recruitment: Marketing tominorities. Psychology and Marketing, 17: 235-255.; Thomas, K.M., & Wise, P.G. 1999. Organizational attractiveness and individual differences: Are diverse applicants attracted by different factors? Journal of Business and Psychology,13: 375-390.
103. Janis, I.L. (1972). Victims of groupthink: A psychological study of foreign policy decisions and fiascoes. Boston: Houghton Mifflin Company.
104. Richard, O.C., Barnett, T., Dwyer, S., Chadwick, K. (2004). Cultural diversity in management, firmperformance, and the moderating role of entrepreneurial orientation dimensions. Academy of Management Journal, 47 (2): 255-266.
105. McMahan, G.C., Bell, M.P., & Virick, M. (1998). Strategic human resource management: Employee involvement, diversity, and international issues. Human Resource Management Review, 8 (3): 193-214.
106. Barney, J. (1991). Firm resources and sustained competitive advantage. Journal of Management, 17(1): 99-120.
107. Kauflin, J. (Jan 23 2018). America’s best employers for diversity. Forbes. Retrieved from https://www.forbes.com/sites/jeffkau.../#84f151c71647
108. Graduate Management Admission Council. (Oct 6 2016Where are women in graduate business school? Retrieved from www.gmac.com/market-intellig...ss-school.aspx
109. Cox, T. H., Lobel, S. A., & McLeod, P. L. (1991). Effects of ethnic group cultural differences on cooperative and competitive behavior on a group task. Academy of management journal, 34(4), 827-847.
110. Richard, O.C., Barnett, T., Dwyer, S., Chadwick, K. (2004). Cultural diversity in management, firm performance, and the moderating role of entrepreneurial orientation dimensions. Academy of Management Journal, 47 (2): 255-266.
111. Dezso, C.L., & Ross, D.G. (2012). Does female representation in top management improve firm performance? A panel data investigation. Strategic Management Journal, 33: 1072-1089.


8.2: The Resolvent
The Transfer Function
One means by which to come to grips with \(R(s)\) is to treat it as the matrix analog of the scalar function
\[\frac{1}{s-b} \nonumber\]
This function is a scaled version of the even simpler function \(\frac{1}{1-z}\) This latter function satisfies the identity (just multiply across by \(1-z\) to check it)
\[\frac{1}{1-z} = 1+z+z^2+\cdots+z^{n-1}+\frac{z^n}{1-z} \nonumber\]
for each positive integer n. Furthermore, if \(|z| < 1\) then \(z^{n} \rightarrow 0\) as \(n \rightarrow \infty\) and so Equation becomes, in the limit,
\[\frac{1}{1-z} = \sum_{n = 0}^{\infty} z^n \nonumber\]
the familiar geometric series. Returning to Equation we write
\[\frac{1}{s-b} = \frac{\frac{1}{s}}{1-\frac{b}{s}} = \frac{1}{s}+\frac{b}{s^2}+ \cdots +\frac{b^{n-1}}{s^n}+\frac{b^n}{s^n} \frac{1}{s-b} \nonumber\]
and hence, so long as \(|s|>|b|\) we find,
\[\frac{1}{s-b} = \frac{1}{s} \sum_{n = 0}^{\infty} \left(\frac{b}{s}\right)^{n} \nonumber\]
This same line of reasoning may be applied in the matrix case. That is,
\[(sI-B)^{-1} = s^{-1} \left(I-\frac{B}{s}\right)^{-1} \left(\frac{1}{s}+\frac{B}{s^2}+ \cdots +\frac{B^{n-1}}{s^n}+\frac{B^n}{s^n}(sI-B)^{-1}\right) \nonumber\]
and hence, so long as \(|s| > ||B||\) where \(||B||\) is the magnitude of the largest element of \(B\)
\[\frac{1}{sI-B} = s^{-1} \sum_{n = 0}^{\infty} \left(\frac{B}{s}\right)^{n} \nonumber\]
Although Equation is indeed a formula for the transfer function you may, regarding computation, not find it any more attractive than the Gauss-Jordan method. We view Equation however as an analytical rather than computational tool. More precisely, it facilitates the computation of integrals of \(R(s)\). However, \(C_{\rho}\) is the circle of radius \(\rho\) centered at the origin and \(\rho > ||B||\) then
\[\begin{align*} \int_{C_{\rho}} (sI-B)^{-1} ds &= \sum_{n = 0}^{\infty} B^n \int_{C_{\rho}} s^{-1-n} ds \\[4pt] &= 2 \pi i I \end{align*}\]
This result is essential to our study of the eigenvalue problem. As are the two resolvent identities. Regarding the first we deduce from the simple observation
\[(s_{2}I-B)^{-1}-(s_{1}I-B)^{-1} = (s_{2}I-B)^{-1}(s_{1}I-B-s_{2}I+B)(s_{1}I-B)^{-1} \nonumber\]
that
\[R(s_{2})-R(s_{1}) = (s_{1}-s_{2})R(s_{2})R(s_{1}) \nonumber\]
The second identity is simply a rewriting of
\[(sI-B)(sI-B)^{-1} = (sI-B)^{-1}(sI-B) = I \nonumber\]
namely,
\[\begin{align*}BR(s) &=R(s)B \\[4pt] &= sR(s)-I \end{align*}\]